Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $q = \dfrac{p - 10}{p^2 - 3p - 70} \times \dfrac{3p^2 + 30p + 63}{p + 9} $
Answer: First factor out any common factors. $q = \dfrac{p - 10}{p^2 - 3p - 70} \times \dfrac{3(p^2 + 10p + 21)}{p + 9} $ Then factor the quadratic expressions. $q = \dfrac {p - 10} {(p + 7)(p - 10)} \times \dfrac {3(p + 7)(p + 3)} {p + 9} $ Then multiply the two numerators and multiply the two denominators. $q = \dfrac {(p - 10) \times 3(p + 7)(p + 3) } { (p + 7)(p - 10) \times (p + 9)} $ $q = \dfrac {3(p + 7)(p + 3)(p - 10)} {(p + 7)(p - 10)(p + 9)} $ Notice that $(p + 7)$ and $(p - 10)$ appear in both the numerator and denominator so we can cancel them. $q = \dfrac {3\cancel{(p + 7)}(p + 3)(p - 10)} {\cancel{(p + 7)}(p - 10)(p + 9)} $ We are dividing by $p + 7$ , so $p + 7 \neq 0$ Therefore, $p \neq -7$ $q = \dfrac {3\cancel{(p + 7)}(p + 3)\cancel{(p - 10)}} {\cancel{(p + 7)}\cancel{(p - 10)}(p + 9)} $ We are dividing by $p - 10$ , so $p - 10 \neq 0$ Therefore, $p \neq 10$ $q = \dfrac {3(p + 3)} {p + 9} $ $ q = \dfrac{3(p + 3)}{p + 9}; p \neq -7; p \neq 10 $